Why not use the MSE instead of the current logistic regression?
The mean squared error (MSE), $J(\theta) = \frac{1}{2m}\sum_{i=1}^m(h_\theta(x_i)-y_i)^2$, is not as appropriate as a cost function for classification, given that the MSE makes assumptions about the data that are not appropriate for classification. Though, as an optimization objective, it is still possible to attempt to minimize MSE even in a classification problem, and thus still learn parameters $\theta$.

The new cost function has better convergence characteristics as it is more inline with the objective.

See link for the precise mathematical formulation that explains these loss functions from a probabilistic perspective. 

Note that the absolute value is redundant because $\forall x:x^2\geq0$.

I hope this clarifies the matter.